matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma

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  10 (*       \ /        This file is distributed under the terms of the       *)
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  14
  15 include "ground_2/relocation/rtmap_id.ma".
  16 include "basic_2/notation/relations/relationstar_4.ma".
  17 include "basic_2/grammar/ceq.ma".
  18 include "basic_2/relocation/lexs.ma".
  19 include "basic_2/static/frees.ma".
  20
  21 (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
  22
  23 definition lfxs (R) (T): relation lenv ≝
  24                 λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≡ f & L1 ⦻*[R, cfull, f] L2.
  25
  26 interpretation "generic extension on referred entries (local environment)"
  27    'RelationStar R T L1 L2 = (lfxs R T L1 L2).
  28
  29 (* Basic properties ***********************************************************)
  30
  31 lemma lfxs_atom: ∀R,I. ⋆ ⦻*[R, ⓪{I}] ⋆.
  32 /3 width=3 by lexs_atom, frees_atom, ex2_intro/
  33 qed.
  34
  35 lemma lfxs_sort: ∀R,I,L1,L2,V1,V2,s.
  36                  L1 ⦻*[R, ⋆s] L2 → L1.ⓑ{I}V1 ⦻*[R, ⋆s] L2.ⓑ{I}V2.
  37 #R #I #L1 #L2 #V1 #V2 #s * /3 width=3 by lexs_push, frees_sort, ex2_intro/
  38 qed.
  39
  40 lemma lfxs_zero: ∀R,I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 →
  41                  R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, #0] L2.ⓑ{I}V2.
  42 #R #I #L1 #L2 #V1 #V2 * /3 width=3 by lexs_next, frees_zero, ex2_intro/
  43 qed.
  44
  45 lemma lfxs_lref: ∀R,I,L1,L2,V1,V2,i.
  46                  L1 ⦻*[R, #i] L2 → L1.ⓑ{I}V1 ⦻*[R, #⫯i] L2.ⓑ{I}V2.
  47 #R #I #L1 #L2 #V1 #V2 #i * /3 width=3 by lexs_push, frees_lref, ex2_intro/
  48 qed.
  49
  50 lemma lfxs_gref: ∀R,I,L1,L2,V1,V2,l.
  51                  L1 ⦻*[R, §l] L2 → L1.ⓑ{I}V1 ⦻*[R, §l] L2.ⓑ{I}V2.
  52 #R #I #L1 #L2 #V1 #V2 #l * /3 width=3 by lexs_push, frees_gref, ex2_intro/
  53 qed.
  54
  55 lemma lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
  56                ∀L1,L2,T. L1 ⦻*[R1, T] L2 → L1 ⦻*[R2, T] L2.
  57 #R1 #R2 #HR #L1 #L2 #T * /4 width=7 by lexs_co, ex2_intro/
  58 qed-.
  59
  60 (* Basic inversion lemmas ***************************************************)
  61
  62 lemma lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻*[R, ⓪{I}] Y2 → Y2 = ⋆.
  63 #R #I #Y2 * /2 width=4 by lexs_inv_atom1/
  64 qed-.
  65
  66 lemma lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻*[R, ⓪{I}] ⋆ → Y1 = ⋆.
  67 #R #I #Y1 * /2 width=4 by lexs_inv_atom2/
  68 qed-.
  69
  70 lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 →
  71                      (Y1 = ⋆ ∧ Y2 = ⋆) ∨
  72                      ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
  73                                       Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
  74 #R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 *
  75 [ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
  76 | #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_next1_aux … H2 … HY1 Hg) -H2 -Hg
  77   /4 width=9 by ex4_5_intro, ex2_intro, or_intror/
  78 ]
  79 qed-.
  80
  81 lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 →
  82                      (Y1 = ⋆ ∧ Y2 = ⋆) ∨
  83                      ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 &
  84                                       Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
  85 #R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 *
  86 [ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
  87 | #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_push1_aux … H2 … HY1 Hg) -H2 -Hg
  88   /4 width=8 by ex3_5_intro, ex2_intro, or_intror/
  89 ]
  90 qed-.
  91
  92 lemma lfxs_inv_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
  93                      L1 ⦻*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
  94 #R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
  95 /6 width=6 by sle_lexs_trans, lexs_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
  96 qed-.
  97
  98 lemma lfxs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 →
  99                      L1 ⦻*[R, V] L2 ∧ L1 ⦻*[R, T] L2.
 100 #R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf
 101 /5 width=6 by sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
 102 qed-.
 103
 104 (* Advanced inversion lemmas ************************************************)
 105
 106 lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 →
 107                              ∃∃L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
 108                                       Y2 = L2.ⓑ{I}V2.
 109 #R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H *
 110 [ #H destruct
 111 | #J #Y1 #L2 #X1 #V2 #HV1 #HV12 #H1 #H2 destruct
 112   /2 width=5 by ex3_2_intro/
 113 ]
 114 qed-.
 115
 116 lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 →
 117                              ∃∃L1,V1. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
 118                                       Y1 = L1.ⓑ{I}V1.
 119 #R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H *
 120 [ #_ #H destruct
 121 | #J #L1 #Y2 #V1 #X2 #HV1 #HV12 #H1 #H2 destruct
 122   /2 width=5 by ex3_2_intro/
 123 ]
 124 qed-.
 125
 126 lemma lfxs_inv_lref_pair_sn: ∀R,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ⦻*[R, #⫯i] Y2 →
 127                              ∃∃L2,V2. L1 ⦻*[R, #i] L2 & Y2 = L2.ⓑ{I}V2.
 128 #R #I #Y2 #L1 #V1 #i #H elim (lfxs_inv_lref … H) -H *
 129 [ #H destruct
 130 | #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 131 ]
 132 qed-.
 133
 134 lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 →
 135                              ∃∃L1,V1. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1.
 136 #R #I #Y1 #L2 #V2 #i #H elim (lfxs_inv_lref … H) -H *
 137 [ #_ #H destruct
 138 | #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 139 ]
 140 qed-.
 141
 142 (* Basic forward lemmas *****************************************************)
 143
 144 lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2.
 145 #R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf
 146 /4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
 147 qed-.
 148
 149 lemma lfxs_fwd_bind_dx: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 →
 150                         R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
 151 #R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV //
 152 qed-.
 153
 154 lemma lfxs_fwd_flat_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, V] L2.
 155 #R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
 156 qed-.
 157
 158 lemma lfxs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, T] L2.
 159 #R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
 160 qed-.
 161
 162 lemma lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ②{I}V.T] L2 → L1 ⦻*[R, V] L2.
 163 #R * /2 width=4 by lfxs_fwd_flat_sn, lfxs_fwd_bind_sn/
 164 qed-.
 165
 166 (* Basic_2A1: removed theorems 24:
 167               llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref
 168               llpx_sn_bind llpx_sn_flat
 169               llpx_sn_inv_bind llpx_sn_inv_flat
 170               llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length
 171               llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx
 172               llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co
 173               llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx
 174 *)